Harmonic progression (HP) – find the 10th term: The second term of an HP is 1/6 and the fourth term is 1/14. Find the 10th term of the HP.

Introduction / Context:
In a harmonic progression (HP), the reciprocals of the terms form an arithmetic progression (AP). Use this transformation to compute the required term index-wise, then invert to return to HP.

Given Data / Assumptions:

• HP terms: H2 = 1/6, H4 = 1/14.
• Define AP terms A_n = 1/H_n.
• Thus A2 = 6 and A4 = 14.

Concept / Approach:
Let A_n = A + (n − 1)d. Using A2 and A4, solve for A and d; compute A10, then take H10 = 1/A10.

Step-by-Step Solution:
A + d = 6 … (1)A + 3d = 14 … (2)Subtract (2) − (1): 2d = 8 ⇒ d = 4.A = 6 − d = 2.A10 = A + 9d = 2 + 36 = 38 ⇒ H10 = 1/38.

Verification / Alternative check:
Check indices: A2 = 2 + 4 = 6 (OK); A4 = 2 + 12 = 14 (OK). Thus inversion back to HP is consistent.

Why Other Options Are Wrong:
1/32, 1/36, 1/40, 1/30 correspond to different A10 values than the derived 38.

Common Pitfalls:
Forgetting to invert at the end; using n in place of (n − 1) inside the AP term formula.

Final Answer:
1/38